To expand search, see Geometry. Laterally related topics: Symmetry, Analytic Geometry, Trigonometry, Pattern, Geometric Theorems, The Pyramid, Similarity, The Triangle, The Method of Exhaustion, Projective Geometry, Algebraic Geometry, Non-Euclidean Geometry, The Parallel Postulate, The Regular Solids, Irrationals, The Pentagram, The Sphere, The Conic Sections, Polygons, Spirals, Line-Point Duality, Geometric Fixed Point Principles, The Cycloid, Tilings, and The Square.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Gerdes, Paulus. On mathematics in the history of sub-Saharan Africa. Historia Math. 21 (1994), no. 3, 345--376. SC: 01A13, MR: 95f:01003.
This paper broadly surveys the recent research in sub-Saharan mathematics (and some related areas as well). Areas discussed include prehistoric mathematics (e.g., the Ishango and Border Cave bones), number systems and symbolism (including algorithms and education), games and puzzles (for example, a leopard-goat-cassava leaf river crossing problem and a "topological" puzzle), symmetry in African art, graphs or networks (e.g. Tschokwe sand drawings), architecture (one case involving magic squares; also a brief reference to fractals). Gerdes mentions string figures as a possibly productive future research area; he gives some starting points. He also discusses related areas, such as technology, and studies on language and mathematical concepts. A goal of the studies mentioned is apparently to better understand mathematics learning in Africa. Some studies focus on logic. Questions on interaction with ancient Egypt are still largely open. A better understanding of Islamic mathematics in sub-Saharan Africa is desirable as well. The author also touches on factors connected with the slave trade; e.g., the remarkable but not perhaps entirely atypical abilities of Thomas Fuller. Includes an extensive bibliography. Closely related topics: Sub-Saharan Africa, TallySystems, Games, Puzzles, Symmetry, Continuous Tracing Problems, Architecture, Magic Squares, Fractals in Art, String Figures, Ancient Egypt, The Reckoning of Time, Education, Mathematics in Language, Logic, The Islamic World, and Thomas Fuller (1710-1790).Modify notes on this entry Modify bibliography entry Make comment on this entry
Kemp, Martin. Spirals of life: D'Arcy Thompson and Theodore Cook, with Leonardo and Dürer in retrospect. Physis Riv. Internaz. Storia Sci. (N.S.) 32 (1995), no. 1, 37--54. SC: 01A99 (92-03), MR: 96j:01047.
Discusses theories of how art appears in biology. The author starts with St. Augustine, who concluded "If, then, we argue from the facts, first, that as everyone admits, not a single visible organ of the body serving a definite function is lacking in beauty, and, second, that there are some parts which have beauty and no apparent function, it follows, I think, that in the creation of the human body God put form before function." The author then discusses and compares the investigations of D'Arcy Thompson and Theodore Cook into the mathematical/biological manifestations of the spiral. Thompson and Cook agreed on many issues, though Thompson didn't approve of the "mystical conceptions" that he found in Cook's work. Specific topics discussed include the appearance of the golden ratio in biological systems (often in the guise of the Fibonacci series), turbulence, and transformations that take one biological object into a related one (one of Thompson's examples compares the skulls of Hyrachyus agrarius and Aceratherium tridactylum). In the process, the author touches on the work of Albrecht Dürer and Leonardo da Vinci (as the title suggests). Obviously, this article can not to be comprehensive, and the author himself tells us that the article is itself intended as a preface; it serves this function well. Both Thompson and Cook were well aware of the mathematical difficulties involved in thoroughly understanding the phenomena they wrote of. Cook wrote "It would only be possible to imagine life or beauty as being 'strictly' mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them." And Thompson wrote "And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies beyond our powers of analysis." The author writes that Thompson ended his book with "a plea for biological mathematicians and mathematical biologists to cultivate 'a field which few have entered and no man has explored'". He continues "Thompson's plea did not fall upon deaf ears, but it is only recently that new techniques of computer modeling have begun to realize something of the potential of some of his techniques." Closely related topics: Art, Biology, Spirals, Proportion and the Golden Ratio, Albrecht Dürer, and Leonardo da Vinci (1452-1519).Modify notes on this entry Modify bibliography entry Make comment on this entry
Phillips, Anthony. The topology of Roman mazes. The Visual Mind, 65--73, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
In this fascinating article, the author analyzes Roman mosaic mazes topologically and uses his conclusions to suggest some reconstructions for a number of damaged Roman mazes. His research allows him to conclude that all mazes occurring in antiquity are meander mazes; the exceptions appear to be because of faulty restoration or recording (p. 66). Roman mazes generally appear to be made of copies (usually four) of identical submazes (he calls this the "standard scheme"). The last of the copies is occasionally varied so that the side holding the entrance gate would only have one path towards the center. Otherwise the standard scheme would dictate that there are two paths running on the side with the entrance gate but only one for sides between the other components. The author calls this variation "the Pompeian Variation", and it seems to be well standardized. The last submaze apparently varies in a fairly standardized way. The submazes themselves are commonly made up of stacks of elementary submazes gamma4, although other cases also occur; the author includes a table listing the submazes and the number of examples from among the Roman mazes that are sufficiently well preserved to be intelligible. The author's systematic treatment makes his proposed restorations seem very plausible. He notes that the basic ideals for the Roman maze seems to have originated in Crete, where there is a famous association between Crete and the legend of the Minotaur. Most significantly, Phillips suggests that the Cretans had an understanding of the topological structure of their mazes: "The cons of Knossos bear at least two other designs relevant to this study. One [K:50] is the four-level maze with level sequence 0 3 2 1 4. It appears on a coin dated circa 431-350 B.C. and is evidence that the Cretans had gone beyond the labyrinth game to analyze the structure of the Cretan maze, because in fact the Cretan maze can be realized as two copies of 0 3 2 1 4, one nested inside the other." A fine article, highly recommended. Closely related topics: The Roman Empire, Mazes, and Greece.Modify notes on this entry Modify bibliography entry Make comment on this entry
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